Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427428 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15369 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,7,20)(3,12,14,24,15,8,21,23)(4,16,17,25,19,31,18,9)(13,27,29,26,30,28,32,22);(1,2)(3,12)(4,16)(5,20)(6,11)(7,10)(8,15)(9,17)(13,28)(14,23)(18,25)(19,31)(21,24)(22,32)(26,29)(27,30);(1,3)(2,8)(4,13)(5,21)(6,15)(7,14)(9,22)(10,24)(11,12)(16,28)(17,32)(18,29)(19,30)(20,23)(25,26)(27,31);(1,4)(2,9,11,25)(3,13)(5,17)(6,19)(7,18)(8,22,12,26)(10,16,20,31)(14,29)(15,30)(21,32)(23,27,24,28)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338619, #72338621, #72338630 |
| maximal normal subgroup of | #72338619, #72338621, #72338630 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334624);